Learn English – Linguistic name for verbs like ‘want’, ‘expect’, ‘beg’,

As Peter Shor indicates in his comment, these are modal verbs and are found in other Germanic languages as well. Their unusual conjugation is due to their historical source:

Germanic modal verbs are preterite-present verbs, which means that their present tense has the form of a vocalic preterite. This is the source of the vowel alternation between singular and plural in German and Dutch. Because of their preterite origins, modal verbs also lack the suffix (-s in modern English, -t in German and Dutch) that would normally mark the third person singular form.

Preterite-present verbs are a very old group of verbs found in Germanic languages that in turn descended from a set of Proto-Indo-European verbs. Originally these were perfect-aspect verbs that referred to a present state.

The common characteristic is that the "present tense" of these verbs was conjugated like the "past tense" (essentially). In Modern English this now means that the present-tense "-s" ending is not used, and that modifiers like "to", "will", etc. can't be added. It happens that these verbs are all modals in Modern English, which probably helps preserve the unusual conjugation.

So this is a bit of a historical accident that made a lot more sense in PIE, and now is frozen that way.

Bare infinitive are placed after certain verbs like "please, let, can, bid, dare," and certain verbs of perception like see, hear etc. Auxiliaries are also used in the same fashion. However, they don't create the impression as in the post.

The examples here are found more in spoken English than in the written English. That day is not far off when the usage referred to would earn a name in grammatical vocabulary. Until such time we are more inclined to call such usage a 'kind of invitation/ command/ request' very much in the nature of imperative mood; only difference is that two sentences are joined without any linker, not even a marker and not confirming to any known rules.